Continuity on Closed and Half-Closed Intervals Exercises

Example 1

What must be true for a function to be continuous on an interval of the form (a,b]?

Answer

Answer. We tweak the definition of continuous to involve the left-sided limit at x = b. f must be continuous on (a,b) and

f(b) must exist.

must exist.

f(b) must equal .

Example 2

What must be true for a function to be continuous on an interval of the form [a,b]?

Answer

f must be continuous on both [a,b) and on (a,b]. Equivalently, f must be continuous on (a,b) and

f(a) and f(b) must exist

and must exist

f(a) must equal and f(b) must equal

To think of this with more words and fewer expressions, the value of f at each endpoint must be what we'd expect if we let x approach the endpoint from within the interval.

Example 3

Determine if f is continuous on each of the following intervals.

[a,b]

[a,b)

(a,b]

(a,b)

Answer

[a,b] (No, because f(a) and f(b) are undefined)

[a,b) (No, because f(a) is undefined)

(a,b] (No, because f(b) is undefined)

(a,b) (Yes)

Example 4

Determine if f is continuous on each of the following intervals.

[a,b]

[a,b)

(a,b]

(a,b)

Answer

[a,b] (No, because f(b) is undefined)

[a,b) (Yes, because and f is continuous on (a,b))

(a,b] (No, because f(b) is undefined)

(a,b) (Yes)

Example 5

Determine if f is continuous on each of the following intervals.

[a,b]

[a,b)

(a,b]

(a,b)

Answer

No to all, because f is not continuous on the open interval (a,b).

Example 6

Determine if the function

is continuous on each of the following intervals.

[-1,0]

[-1,0)

(-1,0]

(-1,0)

[0,1]

[0,1)

(0,1]

(0,1)

Answer

If we draw the function, we see this:

This makes it easier to see what's going on. Since f(0) disagrees with both and , f will be discontinuous on any interval containing 0, and continuous on any other interval.

[-1,0] - No

[-1,0) - Yes

(-1,0] - No

(-1,0) - Yes

[0,1] - No

[0,1) - No

(0,1] - Yes

(0,1) - Yes

Example 7

Determine if the function

is continuous on each of the following intervals.

[0,1]

[0,1)

(0,1]

(0,1)

[1,5]

[1,5)

(1,5]

(1,5)

Answer

Answer. If we graph the function, we see this:

Although f is discontinuous at 1 when we look at the whole graph, f(1) agrees with its right-sided limit (that is, ). This means if x = 1 is the left endpoint of an interval, f can be continuous on that interval.